Remarks on holographic models of the Kerr-AdS$_{5}$ geometry

We study the low-temperature limit of scalar perturbations of the Kerr-AdS$_{5}$ black-hole for generic rotational parameters. We motivate the study by considering real-time holography of small black hole backgrounds. Using the isomonodromic technique, we show that corrections to the extremal limit can be encoded in the monodromy parameters of the Painlev\'e V transcendent, whose expansion is given in terms of irregular chiral conformal blocks. After discussing the contribution of the intermediate states to the quasi-normal modes, we perform a numerical analysis of the low-lying frequencies. We find that the fundamental mode is perturbatively stable at low temperatures for small black holes and that excited perturbations are superradiant, as expected from thermodynamical considerations. We close by considering the holographic interpretation of the unstable modes and the decaying process.